$$ \lim _ { x \rightarrow \frac { \pi } { r } } \frac { \cos x } { \sqrt { 1 + \cot x } - \sqrt { 1 - \cot x } } $$

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To evaluate the limit

$$
\lim _ { x \rightarrow \frac { \pi } { r } } \frac { \cos x } { \sqrt { 1 + \cot x } - \sqrt { 1 - \cot x } },
$$

we need to analyze the behavior of the numerator and denominator as $$ x $$ approaches $$ \frac{\pi}{r} $$.

### Step 1: Analyze the numerator
The numerator is $$ \cos x $$. As $$ x $$ approaches $$ \frac{\pi}{r} $$, we need to determine the value of $$ \cos\left(\frac{\pi}{r}\right) $$.

### Step 2: Analyze the denominator
The denominator is $$ \sqrt{1 + \cot x} - \sqrt{1 - \cot x} $$. We need to evaluate $$ \cot x $$ as $$ x $$ approaches $$ \frac{\pi}{r} $$.

Recall that $$ \cot x = \frac{\cos x}{\sin x} $$. As $$ x $$ approaches $$ \frac{\pi}{r} $$, we need to check if $$ \sin\left(\frac{\pi}{r}\right) $$ is zero or not. If $$ \sin\left(\frac{\pi}{r}\right) \neq 0 $$, then $$ \cot\left(\frac{\pi}{r}\right) $$ will be finite.

### Step 3: Simplifying the denominator
If $$ \cot x $$ approaches a finite value, we can substitute it into the denominator:

$$
\sqrt{1 + \cot x} - \sqrt{1 - \cot x}.
$$

To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator:

$$
\frac{\cos x \left( \sqrt{1 + \cot x} + \sqrt{1 - \cot x} \right)}{(\sqrt{1 + \cot x} - \sqrt{1 - \cot x})(\sqrt{1 + \cot x} + \sqrt{1 - \cot x})}.
$$

The denominator simplifies to:

$$
(1 + \cot x) - (1 - \cot x) = 2 \cot x.
$$

### Step 4: Final limit expression
Thus, the limit can be rewritten as:

$$
\lim _ { x \rightarrow \frac { \pi } { r } } \frac{\cos x \left( \sqrt{1 + \cot x} + \sqrt{1 - \cot x} \right)}{2 \cot x}.
$$

Now we can evaluate this limit by substituting $$ x = \frac{\pi}{r} $$ into the expression.

### Step 5: Evaluate the limit
Assuming $$ \sin\left(\frac{\pi}{r}\right) \neq 0 $$:

1. Calculate $$ \cos\left(\frac{\pi}{r}\right) $$.
2. Calculate $$ \cot\left(\frac{\pi}{r}\right) $$.
3. Substitute these values into the limit expression.

Let's proceed with these calculations.

1. $$ \cos\left(\frac{\pi}{r}\right) $$ is straightforward.
2. $$ \cot\left(\frac{\pi}{r}\right) = \frac{\cos\left(\frac{\pi}{r}\right)}{\sin\left(\frac{\pi}{r}\right)} $$.

Now, we can evaluate the limit directly.

If you provide the value of $$ r $$, I can compute the final limit.
The limit evaluates to $$ \frac{\cos\left(\frac{\pi}{r}\right) \left( \sqrt{1 + \cot\left(\frac{\pi}{r}\right)} + \sqrt{1 - \cot\left(\frac{\pi}{r}\right)} \right)}{2 \cot\left(\frac{\pi}{r}\right)} $$.

Upstudy AI Solution

Answer

Solution

Step 1: Analyze the numerator

Step 2: Analyze the denominator

Step 3: Simplifying the denominator

Step 4: Final limit expression

Step 5: Evaluate the limit

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